This summary shows that there is a unified theory for the prediction of occurrences of turbulence and the catastrophic events in the nature. This topic has a big significance to the science and the society. Flow turbulence has been a challenging subject of fluid dynamics for over 100 years since Reynolds (1883)’ famous experiments on pipe flow demonstrating the transition from laminar to turbulent flows, yet it remains an unsolved problem. Thousands of scientists over the world have been ploughing in this field generation by generation. Now, it is clear that the occurrence of turbulence is resulted from the transition of laminar flows under some disturbance condition. The theoretical method for determining the critical condition of the transition is very difficult, even for the simple Poiseuille flow. Lumley and Yaglom (2001) reviewed the advance of 100 years in turbulent researches, they concluded that it is very far from approaching a comprehensive theory and the final resolution of the turbulent problem. Various stability theories emerged during the past 100 years, but few are satisfactory in the explanation of the various flow instabilities and the related complex flow phenomena. Linear stability analysis has been used for this purpose for many years and was successful for certain flows such as Taylor-Couette flow and boundary layer flow on a flat plate, but it is failed to plane and pipe Poiseuille flows.

 

As is well known, the pipe Poiseuille flow (Hagen-Poiseuille) is linearly stable for all the Reynolds number Re by linear eigenvalue analysis (Schmid and Henningson, 2001).  However, experiments showed that the flow would become turbulence if Re (=ρUD/μ) exceeds a value of 2000. Experiments also showed that disturbances in a laminar flow could be carefully avoided or considerably reduced, the onset of turbulence was delayed to Reynolds numbers up to Re=O(10⁵). For Re above 2000, the characteristic of turbulence transition depends on the disturbance amplitude and frequency, below which transition from laminar to turbulent state does not occur regardless of the initial disturbance amplitude. Linear stability analysis of plane parallel flows gives critical Reynolds number Re (=ρVh/μ) of 5772 for plane Poiseuille flow, while experiments show that transition to turbulence occurs at Reynolds number of order 1000. The resolution of these paradoxes is very difficult (Trefethen et al, 1993). Grossmann (2000) commented that this discrepancy demonstrates that nature of the onset-of-turbulence mechanism in this flow must be different from an eigenvalue instability. Although some linear three-dimensional mechanism and some nonlinear stability theories have been proposed to predict the transitional Reynolds number, these theories do not seem to offer a good agreement with the experimental data.

 

There is also an energy method which is used in the prediction of flow instability and turbulence transition. In this method, one observes the rate of increasing of disturbance energy to study the instability of the flow system. It is considered that turbulence shear stress interacts with the velocity gradient and the disturbance gets energy from mean flow in such a way. Thus, the disturbance is amplified and the instability occurs with the energy increasing of disturbance. Therefore, it is recognized that it is the basic state vorticity leading to instability. However, this energy method does not provide a good agreement with experiments either. In recent years, various transition scenarios have been proposed for the subcritical transition. Although we can get a better understanding of the transition process from these scenarios, the mechanism is still not fully understood and the agreement with the experimental data is still not satisfied.

 

Recently, a new mechanism of flow instability and turbulence transition is presented for parallel shear flows and an energy gradient theory of hydrodynamic instability is proposed (Dou, 2003; Dou, 2004). It is stated that the total energy gradient in the transverse direction and that in the streamwise direction of the main flow dominate the disturbance amplification or decay. Thus, they determine the critical condition of instability initiation and flow transition under given disturbance. A new dimensionless parameter K for characterizing flow instability is proposed for wall bounded shear flows, which is expressed as the ratio of the energy gradients in the two directions. It is thought that flow instability should first occur at the position of K reaching its maximum (Kmax) which is the most dangerous position, and then it spreads outside. This theory has been proved to be valid by the experimental data in literature. It is found that the turbulence transition takes place at a critical value of Kmax of about 385 for the both plane Poiseuille flow and pipe Poiseuille flow and 360 for plane Couette flow, below which no turbulence will occur regardless the disturbance. This shows that the transition occurs at a critical value of Kmax of 360-385 for parallel flows. We believe that the energy gradient theory can also be used to other more complex flows, and is a universal theory. Actually, the Rayleigh-Benard convective instability, Kelvin-Helmholtz instability and the stratified flow instability could also be considered as those produced by the energy gradient transverse to the flow.

 

The energy gradient theory can be not only used to predict the generation of turbulence, but also it may be applied to the area of catastrophic event predictions, such as rupture of composite materials, weather forecast, earthquakes, landslides, and mountain coast, etc., and even the microscopic flows in biomechanics. The breakdown of these mechanical systems can be universally described in detail using this theory. In a material system, when the maximum of energy gradient in some direction is greater than a critical value for given material properties, the system will be unstable. If there is a disturbance input to this system, the energy gradient may amplify the disturbance and lead to the system breakdown.  Sornette (2002) also described a unifying approach for modeling and predicting these catastrophic events or ‘‘ruptures,’’ that is, sudden transitions from a quiescent state to a crisis. He stated that such ruptures involve interactions between structures at many different scales. He believes that it is possible to develop universal theory and tool for predicting these catastrophic events.  The energy gradient theory (Dou, 2003) describes a mechanism of failure of mechanical system in a special way. It will demonstrate its powerfulness in the application of catastrophic predictions in near future.

 

References:

1.    O. Reynolds, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels, Phil. Trans. Roy. Soc. London A, 174, 935-982 (1883).

2.    J.L.Lumley and A.M. Yaglom, A Century of Turbulence, Flow, Turbulence and Combustion, 66, 241-286 (2001).

3.    P.J.Schmid, and D.S.Henningson, Stability and transition in shear flows, (Springer, New York, 2001).

4.    L.N.Trefethen, A.E. Trefethen, S.C.Reddy, T.A.Driscoll, Hydrodynamic stability without eigenvalues, Science, 261, 578-684 (1993).

5.    S.Grossmann, The onset of shear flow turbulence, Reviews of modern physics, 72, 603-618 (2000).

6.    H.-S. Dou, Energy Gradient Theory of Hydrodynamic Instability, submitted  (2003).

7.    H.-S. Dou, Mechanism of bypass transition in wall shear flows, Accepted to be presented at The 4th International conference on Fluid Mechanics, July, 2004, Dalian, China.

8.    D. Sornette, Predictability of catastrophic events: Material rupture, earthquakes, turbulence, financial crashes, and human birth, PNAS, 99, 2522-2529 (2002).